Rotational Motion: Momentum Conservation

[Prannoy Mehta]

Homework Statement

A solid sphere is set into motion on a rough horizontal surface with a linear speed v in the forward direction and angular speed v/r in the anticlockwise direction. Find the linear speed of the sphere when:

a) When it stops rotating
b) when slipping ceases

Homework Equations

Basic Angular Momentum based equations

The Attempt at a Solution

Now I imagined it moving as described. Since the floor is rough a torque will be acted by the floor which acts in the backward direction. Causing the angular velocity to reduce. The velocity of the centre of mass remains intact. So applying angular momentum for case (i) Where v' is the velocity of the centre of mass.

a) mvr + 2/5 mrv = mv'r (Iw = 2/5mvr)
v' = 7v/5

Now the friction acts to the torque to increase the angular velocity in the clockwise direction till pure rolling is achieved. So now when achieved the sphere is rolling clockwise, and the velocity remains in the positive direction.

b) mvr + 2/5 mrv = mv'r - 2/5mv'r (Iw = 2/5mvr)
v' = 7v/3

The answer for (a) is 3v/5 and (b) is 3v/7. If you could explain my mistakes it would be very helpful. Thanks in advance.

 

[TSny]

You have some sign errors in your equations for (a) and (b). It might help to state the location of the point that you are choosing as the origin for the purpose of calculating the angular momentum.

 

[Prannoy Mehta]

You have some sign errors in your equations for (a) and (b). It might help to state the location of the point that you are choosing as the origin for the purpose of calculating the angular momentum.

I don't get it. My origin can be taken as any fixed point which lies at the left of the sphere during the shifting of anticlockwise to clockwise rotation.

Is my concept flawed or are my sign convention. Could you explain why they are wrong?

 

[TSny]

I don't get it. My origin can be taken as any fixed point which lies at the left of the sphere during the shifting of anticlockwise to clockwise rotation.

Do you mean any fixed point on the floor along the line of motion? If so, OK.

Is my concept flawed or are my sign convention. Could you explain why they are wrong?

For the initial time, how would you describe the direction of the angular momentum due to the linear motion of the center of mass of the sphere? How would you describe the direction of the angular momentum due to the rotation of the sphere about the center of mass?

 

[Prannoy Mehta]

Yes, that's what I meant.

Oh well, initially the ball is simply set into motion. With it rotating anticlockwise the velocity is perpendicular (Outside the page or + z axis, if a 2D diagram of the sphere is drawn as suggested) to the velocity of the centre of mass being in the +x axis. I guess that's what you asked :/. Anyhow, in the second case the velocity is perpendicular (Into the page -z axis) to the velocity of the centre of mass being in the +x axis as earlier.

I am not sure if this is what you wished to know :/

 

[TSny]

I'm not sure if you have covered the vector definition of angular momentum. If so, you should be able to specify the direction of the angular momentum vector associated with the linear motion of the center of mass of the sphere.

But, we can avoid that and just use clockwise and anticlockwise for directions of angular momentum. So, my question is the following. What is the direction of the angular momentum term mvr [associated with the linear motion v of the sphere]? Is it clockwise or anticlockwise?

 

[faradayscat]

Now I imagined it moving as described. Since the floor is rough a torque will be acted by the floor which acts in the backward direction. Causing the angular velocity to reduce.

Think about it, in what direction should the torque act as to produce a counter-clockwise angular velocity?

 

[Prannoy Mehta]

mvr is the angular momentum of the entire sphere (assumed to sum up to one point) from the point where it touches the ground. mvr is clockwise and I just got my answer. Thank you so much. It was very silly of me. Thank you.

I have not yet understood the part where angular velocity is a vector. I know it is there as the text recommended mentions a formulae v (vector) = w ( vector) x r(vector). Could you suggest me something I can comprehend and which could increase the grasp of the concept? I don't want to dig in too deep right now but would also like to learn.

 

[Prannoy Mehta]

Think about it, in what direction should the torque act as to produce a counter-clockwise angular velocity?

The sphere is already moving counter clockwise. The frictional force will go against this rolling motion as the bottom of the sphere tends to move with a velocity v unlike perfect rolling. Friction will oppose the change and hence create a torque in the clockwise direction. Hence the torque reduces the angular velocity. (It will become negative and stop after perfect rolling is achieved. )

 

[TSny]

mvr is the angular momentum of the entire sphere (assumed to sum up to one point) from the point where it touches the ground. mvr is clockwise and I just got my answer.

Good!

I have not yet understood the part where angular velocity is a vector. I know it is there as the text recommended mentions a formulae v (vector) = w ( vector) x r(vector). Could you suggest me something I can comprehend and which could increase the grasp of the concept? I don't want to dig in too deep right now but would also like to learn.

This is probably not the place to try to give a mini-lecture on the vector nature of angular velocity and angular momentum. However, if you wish to study what your text has on this topic and you then have some questions, you can post your questions here at the forum (starting a new thread).

 

[faradayscat]

The sphere is already moving counter clockwise. The frictional force will go against this rolling motion as the bottom of the sphere tends to move with a velocity v unlike perfect rolling. Friction will oppose the change and hence create a torque in the clockwise direction. Hence the torque reduces the angular velocity. (It will become negative and stop after perfect rolling is achieved. )

Ah, my bad, you're right!

 

[Prannoy Mehta]

Good!

This is probably not the place to try to give a mini-lecture on the vector nature of angular velocity and angular momentum. However, if you wish to study what your text has on this topic and you then have some questions, you can post your questions here at the forum (starting a new thread).

Ah, fine thank you. I will post in the forums if I have any further queries.